Part 1 of Sklar's Theorem finds much application in statistical dependency analysis [1]}. In contrast to classical dependency measures, such as Pearson correlation, copulae are a more flexible tool that allow the decoupling of the marginals and the dependency structure. Part 2 of Sklar's Theorem is of relevance for statistical modeling, and more precisely, to define multivariate distributions. Given marginal distributions, which are typically much easier to estimate than the full joint distribution, and a copula \(C\)  we can “couple” the marginals and the dependency structure to a multivariate joint distribution.
This perspective finds various applications in the context of finance and related disciplines that need to take heavy tails and tail dependencies into account, see [2]} for an overview.
In Section  of the Appendix we give some illustrative examples and further details about properties of copula distributions.
